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Fractional Differential Equations
- Publisher
- Springer Journals
- Copyright
- Copyright © 2012 by Springer Science+Business Media B.V.
- Subject
- Mathematics; Theoretical, Mathematical and Computational Physics; Computer Science, general; Mathematics, general; Statistical Physics, Dynamical Systems and Complexity; Mechanics
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-012-9679-1
- Publisher site
- See Article on Publisher Site
Abstract
In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument $$\left \{\begin{array}{l@{\quad}l}D_{0^+}^{\alpha}u(t)+h(t)f(u(\theta(t)))=0, & t\in ( 0,1 ),\ n-1<\alpha\leq n,\\[3pt]u^{(i)}(0)=0, & i=0,1,2,\ldots,n-2,\\[3pt][D_{0^+}^{\beta} u(t)]_{t=1}=0, & 1\leq\beta\leq n-2,\end{array} \right .$$ where n>3 (n∈ℕ), $D_{0^{+}}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order α,f:[0,∞)→[0,∞), h(t):[0,1]→(0,∞) and θ:(0,1)→(0,1] are continuous functions. Some novel sufficient conditions are obtained for the existence of at least one or two positive solutions by using the Krasnosel’skii’s fixed point theorem, and some other new sufficient conditions are derived for the existence of at least triple positive solutions by using the fixed point theorems developed by Leggett and Williams etc. In particular, the existence of at least n or 2n−1 distinct positive solutions is established by using the solution intervals and local properties. From the viewpoint of applications, two examples are given to illustrate the effectiveness of our results.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Feb 8, 2012